The close-packed crystal space matrix of 12 rays, 4 planes, delineating Tetrahedron and octahedron edges, is known in classic chemistry as the face-centered-cubic or cubic close-packed crystal structure. In fact, this atomic packing matrix positions atoms at the vertices and center of the cubeoctahedron, Buckminster Fuller’s beloved Tensegrity structure, or vector equilibrium, defining the planes of cubic, tetrahedral and octahedral space.The best way to stiffen a cube of sticks is with diagonals on the 6 faces which connect to make a simpler form, the tetrahedron.Remove 2 of the 6 tetrahedral edges, leaving 2 pair of 90 degree oriented edges, which can be dipped into soap solution to form a beautifully balanced minimal surface, the hyperbolic paraboloid saddle curves of balanced curvature at every point.

The 90 degree hyperbolic paraboloid of a regular tetrahedron is most symmetric within the close-packed crystal matrix, which can be expanded to create a curved space tunnel system.

These drawings show how the regular tetrahedron fits into the cube, the hyperbolic paraboloid (hypar) generated from the tetrahedron, and the curved space created (in this case, 6 smoothly-connected tetrahedral hypars within the boundaries of the cubeoctahedron)

Tetrahedral hyperbolic paraboloids can be connected in other configurations to create pleasing forms such as this modular tent design.